Finally, it should be stressed that in the context of a shallow-water model, alternative variables to the (relative) vorticity \(\zeta \)  are the absolute vorticity \(\omega =\mathbf {\nabla }^\perp \mathbf {\cdot u} +f\)  or potential vorticity \(q = (\mathbf {\nabla ^\perp \cdot u} + f)/h\) .
As the potential vorticity is conserved along the flow, it is often preferred to \(\zeta \) , as in the case of the compatible finite element discretisations in [1]} and [2]}.
In particular, the APVM and SUPG stabilisations derived in the aforementioned papers dissipate the enstrophy \(hq^2\) , while conserving the system's total energy. These works also solved the vorticity evolution equation corresponding to the whole shallow-water equation for the velocity (REF ), whereas in this section we consider only the transport part; as mentioned in Section REF  the motivation here is to find a “black box” to solve the vector transport equation.
The addition of the SUPG stabilisation is still consistent within the transport step, and any errors arising from not applying SUPG to the whole equation will do so in the form of a splitting error in time.
Note that this SUPG setup is different to the ones used in [2]} and [4]}.
In the former, a different vorticity variable is used, leading to a forcing contribution of the form \(g{\mathbf {\nabla }{(h+h_b)}}\) , which vanishes in the vorticity evolution equation (since \(\mathbf {\nabla ^\perp \cdot } {\mathbf {\nabla }{(h+h_b)}} = 0\) ).
In the latter, there is no time splitting, and a forcing contribution \(\mathbf {\nabla ^\perp \cdot } \mathbf {J}\)  for some baroclinic forcing terms \(\mathbf {J}\)  is included in the vorticity equation's residual. While these approaches avoid errors due to time splitting and lead to additional conservation properties such as energy conservation, they require additional information from the equations and cannot be used as “black box” vector transport methods.
In particular, this is a drawback for code implementations: while a “black box” setup can be used for a variety of different equation sets, in the specific setups of [2]} and [4]}, the transport implementation has to be adjusted each time the overall equation sets are changed.
